We study the perturbation theory of the angular pair correlation function g(rω1ω2)in a molecular fluid. We consider an anisotropic pair potential of the form u = u0 + ua, where u0 is an isotropic 'reference' potential, and for simplicity in this paper we assume the perturbation potential ua to be 'multipole-like', i.e., to contain no l = 0 spherical harmonics. We expand g in powers of ua about g0, the radial distribution function appropriate to u0. This series is examined by expanding ha = h−h0 (where h = g−1) and its corresponding direct correlation function ca in spherical harmonic components. We consider approximate summations of the series in ua that automatically truncate the corresponding harmonic series, so that the Ornstein–Zernike (OZ) equation relating ha and ca can be solved in closed form. We first expand ca = c1 + c2 + … where cn includes all terms in ca of order (ua)n. Taking ua to be a quadrupole–quadrupole interaction, we find that a 'mean field' (MF) approximation ca = c1 gives rise to only three nonvanishing harmonic components in ha, so that OZ is solved explicitly in Fourier space. The MF solution for multipoles of general order is given in an appendix. Graphical methods are then used to identify the class of all terms in the ua series that are restricted to the harmonic space defined by MF. A portion of this class can be summed by solving OZ with the closure ca = −βg0ua + h0(ha−ca), where β = (kT)−1, h0 = g0−1 This system is designated as generalized MF (GMF), and solved by numerical iteration. Numerical results from MF and GMF are presented for quadrupolar ua, taking u0 to be a Lennard-Jones potential. Symmetries imposed by the restricted harmonic space are foreign to the full g, yet harmonics within this space are sufficient for evaluation of many macroscopic properties. The results are therefore evaluated in harmonic form by comparison with the corresponding harmonic components of the 'correct' g as evaluated by Monte Carlo simulation.
Mean-Field and Perturbation Theory of Vortex-like Composite Fermions
